Question

Express the rational function as a sum or difference of two simpler rational expressions. \(\frac{x}{x^{2}-4}\)

Short answer

\( \frac{x}{x^2-4} = \frac{1/2}{x-2} + \frac{1/2}{x+2} \).

Step-by-step solution

Factor the Denominator

The first step is to factor the denominator of the rational expression. We have \[ x^2 - 4 \] which is a difference of squares. It can be factored as: \[ x^2 - 4 = (x - 2)(x + 2) \]

Decompose into Partial Fractions

Now, express the given rational function \( \frac{x}{x^2 - 4} \) as a sum of two simpler fractions. By partial fraction decomposition, assume: \[ \frac{x}{x^2 - 4} = \frac{A}{x - 2} + \frac{B}{x + 2} \] where \( A \) and \( B \) are constants to be determined.

Clear the Denominator

Multiply both sides of the equation by the common denominator \((x - 2)(x + 2)\) to clear the fractions: \[ x = A(x + 2) + B(x - 2) \]

Expand and Collect Like Terms

Expand the right side of the equation: \[ A(x + 2) + B(x - 2) = Ax + 2A + Bx - 2B \] Collect like terms: \[ (A + B)x + (2A - 2B) \]

Solve for Constants

To find \( A \) and \( B \), equate coefficients from both sides of the equation. 1. For \( x \): \[ A + B = 1 \] 2. For the constant term: \[ 2A - 2B = 0 \] Solve these two equations to find the values of \( A \) and \( B \).

Solve the Equations

From the second equation: \[ 2A - 2B = 0 \Rightarrow A = B \] Substitute \( A = B \) into the first equation: \[ A + A = 1 \Rightarrow 2A = 1 \Rightarrow A = \frac{1}{2} \] Therefore, \( B = \frac{1}{2} \).

Write the Final Expression

Substituting \( A = \frac{1}{2} \) and \( B = \frac{1}{2} \) back into the partial fraction decomposition gives: \[ \frac{x}{x^2 - 4} = \frac{1/2}{x - 2} + \frac{1/2}{x + 2} \]

Key concepts

Rational Expressions

Rational expressions are essentially fractions where both the numerator and denominator are polynomials. Just like you would with regular numbers, you need to ensure that when dealing with rational expressions, the denominator is never zero, as this would make the expression undefined. For example, in the rational expression \( \frac{x}{x^2 - 4} \), the denominator \( x^2 - 4 \) can't be zero. If you solve \((x^2 - 4 = 0)\), you'll find that \(x = -2\) and \(x = 2\) are the values where the expression would be undefined. This step is crucial to avoid mistakes when manipulating rational expressions.
Also, when simplifying or performing operations with these expressions, such as addition, subtraction, multiplication, or division, we often factor the polynomials if possible. This could help us cancel terms and simplify the expression further.

Algebraic Fractions

Algebraic fractions are quite similar to rational expressions and work on the same principle of having polynomials in both the numerator and the denominator. One essential operation you can perform on algebraic fractions is partial fraction decomposition. This method is especially useful when you need to simplify complex fractions into simpler ones, which are often easier to integrate, differentiate, or solve in equations.
Similar to numerical fractions, the rules of arithmetic apply. To add or subtract algebraic fractions, you need a common denominator. For multiplication and division, just like numerical fractions, you multiply across the numerators and denominators, or flip the fraction when dividing.

• To add or subtract: Find the least common denominator (LCD).
• To multiply: Multiply the numerators together, then the denominators.
• To divide: Multiply by the reciprocal of the denominator fraction.

Understanding these basic principles can make working with algebraic fractions much more straightforward.

Factoring Polynomials

Factoring polynomials is a key skill needed when working with rational expressions and algebraic fractions. Factoring involves rewriting a polynomial as a product of simpler polynomials. This process can simplify expressions and is particularly useful in partial fraction decomposition.
A common type of factoring involves recognizing special polynomial structures, like the difference of squares. For example, \( x^2 - 4 \) is a difference of squares and can be factored into \( (x - 2)(x + 2) \).
Steps for Factoring Polynomials:

• Identify if the polynomial is a special case such as a difference of squares, perfect square trinomial, or a sum/difference of cubes.
• For basic trinomials, look for two numbers that multiply to give you the last term and add to give you the middle term.
• Always check if there is a greatest common factor (GCF) that can be factored out first.

Once factored, polynomials can be more easily worked with, especially in simplifying or decomposing rational expressions.